The Bergen Snow Simulator (BESSI) is a surface energy and mass balance model for glaciated regions with a flexible spatial domain. In this study, the domain is Greenland with an equidistant resolution of 10 km. The topography of the ice sheet is based on ETOPO1 and remains fixed throughout the simulations. The vertical dimension consists of up to 15 snow or firn layers that are adjusted by splitting or merging layers depending on the snow mass in each grid cell . The five daily input variables are air temperature and dew point at 2 m above ground, the amount of precipitation, and surface downwelling shortwave and longwave radiation. The top layer changes its mass and energy according to the forcing of the input variables. Precipitation falls as snow when the air temperature is below 0 ∘C and as rain otherwise. Meltwater percolates down into deeper layers and refreezes. Horizontal exchanges of mass or energy are deemed negligible on the 10 km grid. When there is no more snow left to melt, the excess energy is used to melt ice. Corrections are made when the melt exceeds the existing amount of ice (Appendix ). For a detailed description of the snow model, see and . The performance of BESSI has been compared with other SMB models in . Although snowfall and runoff are lower in BESSI than in other SMB models, the SMB and its trend are consistent with most other studied models because both biases cancel each other out.

BESSI uses parameterizations of several physical processes. In this study, we vary the albedo and turbulent heat exchange parameters (Table ), which contribute to the parameter uncertainty discussed below. The albedo changes caused by ageing of the snow are parameterized depending on temperature, whereas the ageing is accelerated at 0 ∘C depending on the liquid water content . Thus, the albedo of the snow can take values between the prescribed albedos of fresh snow and firn. Ice is assigned a separate albedo. The turbulent sensible heat flux depends on the difference between air and surface temperature, and on the turbulent heat exchange coefficient, which is a model parameter describing both changes in wind speed and efficiency of the turbulent exchange .

The model parameters of BESSI are calibrated to the RACMO SMB . Here, we use an ensemble of equally plausible model parameter settings based on a multivariate calibration . For the calibration, BESSI was run for 500 years with ERA-Interim as forcing data using different parameter combinations. The performance of the simulation is compared to the RACMO SMB over the period 1979–2017 on an annual basis. We are using seven measures of goodness of fit, based on the bias, the mean absolute deviation (MAD), and the root-mean-square error (RMSE) of the SMB. The bias is the difference between the ice-sheet-wide integrated SMB between RACMO and BESSI, while we calculate three representations of RMSE and MAD. The first calculates the Greenland-wide SMB and its temporal MAD over the years, the second calculates a temporal MAD for each grid point and averages them over all grid points, with the third and final being the MAD over all points in space and time. A similar approach is used for the RMSE. In total we are using seven objective functions for the multivariate optimization. We evaluate the performance of BESSI relative to all the objective functions. In a non-ideal world not all objectives can be minimized simultaneously. This yields multiple equally plausible optimal solutions, where one objective function can not be improved without compromising another. The ensemble of these optimal solutions is referred to as Pareto optimal set. Similar to the method used by , we calculate the Pareto optimal set. This yields a total of 16 different solutions whose parameter ranges are given in Table .

We use ESM output of CMIP6 for the period of 2015–2100 . The Tier 1 scenarios (with increasing radiative forcing: SSP126, SSP245, SSP370, and SSP585) from ScenarioMIP are selected for this study because they encompass a wide range of future forcing possibilities and are available for many different ESMs. We selected 26 ESMs that provide all of BESSI's input variables for at least two scenarios (Appendix Table ), with the exception of the dew point, which is calculated from the relative humidity if necessary.

The input variables are interpolated linearly to the 10 km BESSI grid. ESM biases are calculated based on the delta method by comparing the daily mean of the historical simulation and the daily mean of the ERA-Interim reanalysis in the period of 1979–2014, in which both datasets are available. For all input variables except precipitation, the differences between the daily means are subtracted from the future projection. These differences also include discrepancies in topography, so the dependence of, for example, temperature on elevation is accounted for in the additive bias correction. As mentioned in Sect. , BESSI uses ETOPO1 and accounts for the differences with the ERA-Interim topography by performing a correction with a constant moist adiabatic lapse rate. Precipitation is bias corrected by the ratio of ERA-Interim and historical mean precipitation because its high variability would lead to negative values if the difference was used. During the winter, shortwave radiation may be very weak so that the bias correction can lead to localized, small negative values. These values are set to zero. The daily means of precipitation are affected by individual intense precipitation events due to the short length of the historical period. The monthly biases are less affected by these events, and therefore we multiply the projected precipitation with the ratio of the monthly mean precipitation of the historical reference and the reanalysis data to perform bias correction instead of the daily means.

Throughout the 21st century, the median air temperature over all ESMs rises in every scenario except in the scenario with the smallest increase in greenhouse gases (SSP126), where it remains almost constant during the second half of the century (Fig. a). While shortwave radiation decreases slightly, precipitation, longwave radiation, and dew point increase over the course of the century (Fig. b–e). The stronger the greenhouse gas forcing, the larger the change in these variables. For each variable except precipitation, there are distinct differences in ESM medians between all scenarios at the end of the century, and the differences between scenarios are of similar magnitude to the interquartile ranges. The trends in precipitation are weaker compared to the ranges of values between the ESMs.

We conduct two different kinds of SMB simulations. (i) In the main ensemble, the forcing data for four climate scenarios are taken from different ESMs, and the snow model parameters are varied. It illustrates the temporal and spatial behaviour of the SMB and it enables us to separate the different uncertainty components. (ii) The “single-forcing” ensemble shows the influence of the individual input variables.

The main ensemble uses 96 selected ESM–scenario combinations (Table ). In addition, we conduct 26 simulations for the historical reference period (1979–2014), i.e. one for each ESM. Each of the simulations is conducted with 16 different snow model parameter sets, resulting in 1952 simulations. The selection process of the parameter combinations is described in Sect. . The firn cover is initialized by forcing BESSI with ERA-Interim reanalysis data for 540 years, to reach a dynamically and thermodynamically stable firn cover at the year 2014. The long response time of the firn cover requires an initialization period of several hundred years, which is realized by forcing the model with the ERA-Interim data 15 times back and forth . For the historical time period, the initialization ends in 1979 after 14 ERA-Interim cycles back and forth. For every parameter set, the same initialized firn cover is used to save computation time, but the bias caused by this inconsistency is generally overcompensated after a few years of climate forcing.

In the single-forcing simulations, the transient ESM simulations are used as input for only one variable, and the daily ERA-Interim climatology is used for the others to assess the influence of each variable on the SMB. The scenario SSP585 is chosen because it is available for all 26 ESMs, and we used the snow model parameter set that produces the best results in the calibration with RACMO (Sect. ). For precipitation, the daily ERA-Interim climatology cannot be used as it overestimates the surface albedo due to unrealistic small amounts of snowfall every day . This leads to an overestimation of the mass balance of up to 40 % . Instead we use the monthly precipitation climatology and distribute the ERA-Interim monthly average PERAim following the temporal distribution of precipitation in the ESM simulation: 1Pyear,climd=Pyeard⋅PERAimPyear,modelm, where P stands for precipitation, “m” stands for monthly mean, “d” stands for daily mean, and year stands for the point in time of the simulation. Therefore, the climatological daily precipitation distribution differs for each ESM, but the monthly averages are identical. For each of the 26 ESMs, we conducted six simulations for the SMB: a reference simulation with the historical climatology and five simulations with different transient variables (air and dew point temperature, precipitation, shortwave and longwave radiation). We need a separate reference simulation for each ESM because the precipitation distribution differs for each ESM according to Eq. ().

In this section, we show temporal and spatial differences between the ESMs and climate scenarios of the median SMB over all parameter combinations. The median SMB at the end of the century over the ESMs and snow model parameters is shown for the different climate scenarios in Table . The surface mass balance decreases relative to the historical simulations in all scenarios (Fig. ). In the moderate scenario SSP126, the SMB is relatively stable to the end of the century. Higher emissions of greenhouse gases (stronger forcing) lead to a lower SMB (SSP245, SSP370, SSP585). With stronger warming, the range in simulated SMB for different ESMs increases, although the range in input variables except precipitation does not seem to depend on the scenario (Fig. ). For precipitation, the interquartile range between the ESMs increases only slightly with stronger greenhouse gas forcing. Precipitation variability alone cannot explain the larger interquartile range in SMB in the warmer scenarios. The reason for the increasingly dissimilar SMBs with stronger greenhouse gas forcing is that larger changes in the input variables have a larger cumulative effect on the SMB (Sect. ).

When BESSI is forced with ERA-Interim data (Fig. , orange), a relatively low SMB in the early 21st century is apparent. This correlates with more frequent Greenland blocking . A similar reduction in SMB is not observed when forcing BESSI with historical ESM data (Fig. , black) because the coarse horizontal resolution hampers the representation of the observed blocking and its increased activity .

Spatial anomalies for the last decade of the SMB in the low-emission scenario SSP126 and the high-emission scenario SSP585 are shown in Fig. . In the west of Greenland, the SMB in the 2090s is lower than in ERA-Interim (1979–2014), independent of the scenario (Fig. a and b). In this region, higher temperatures lead to increased melt. In the centre of the ice sheet, the SMB is slightly higher than in ERA-Interim, especially in the southeast. There, heavier precipitation occurs under a warmer climate. However, the SMB increase in the centre is outweighed by the SMB decrease at the margin of the ice sheet. These SMB changes are much more pronounced in the high-end scenario SSP585 because of the enhanced change in the input variables. Currently observed SMB changes are dominated by amplified melting in the west and by snowfall in the east . In the north, the temperatures are too low for much melt at the present day, but with an average increase of temperature over the ice sheet of approximately 6 K in SSP585 (Fig. a), melt increases considerably there.

At the margin of the ice sheet, the standard deviation of the SMB between the ESMs is largest (Fig. c and d). The relative standard deviation of the SMB reaches the highest values near the equilibrium line (Fig. e and f), and thus the choice of ESM is decisive for the SMB in this region. In the high-emission scenario SSP585, the equilibrium line is subject to substantial uncertainty, which is greater than in the moderate scenario SSP126 (Fig. ). Equilibrium line changes show that the differences between ESMs driven by the same scenario increase with stronger greenhouse gas forcing (Fig. ).

Having examined the spatial variations between ESMs, we next study the variance of the full ensemble containing ESMs, emission scenarios and snow model parameters. We split the variance in spatially integrated SMB over all simulations into four different components using a method based on and described in more detail in Appendix : a fourth degree polynomial fit is applied to the decadal running mean of spatially integrated SMB for each individual simulation to separate trends from variations on small timescales. The residuals of the fits are considered the internal variability of the system, for example fluctuations in SMB caused by alternating dry and wet periods. The law of total variance is applied to the whole ensemble of polynomials to split the total variance into three independent components for each year. These components are the variances caused by ESMs, climate scenarios, and BESSI parameters (albedo of fresh snow and firn, turbulent heat exchange coefficient). These variances quantify three relevant sources of uncertainty, with internal variability being the fourth.

The sum of the different uncertainty components increases strongly over the course of the century (Fig. a). The relative contributions of the different uncertainty components are shown in Fig. b by normalizing with the sum of all components. In the first years of the simulations, the internal variability is the largest source of uncertainty, showing that it is most important in the absence of external forcing. While the scenario uncertainty has the smallest contribution in the beginning, its importance increases in the second half of the century, as decarbonization measures and the adaption of the climate system take time . The parameter uncertainty is slightly larger than the scenario uncertainty at first, but its relative importance decreases over time. Its overall small contribution to uncertainty indicates that the results of our SMB simulations are almost independent of the specific parameter combination of BESSI. The parameter uncertainty does not depict the total snow model uncertainty because the approach to calculate the SMB is the same regardless of the parameter combination, whereas differences in the ESMs are caused by different ways of simulating the processes. The spatial resolution necessarily contributes to the uncertainty in SMB modelling because elevation and associated temperature differences on the sub-grid scale can lead to unrealistically high temperatures prescribed above the ablation zone, reducing the SMB . Furthermore, the calculation of precipitation and runoff is less accurate in BESSI compared to other snow models, and these biases could increase in a warming climate , which is not represented in the parameter uncertainty either.

A few years into the simulation, the ESM uncertainty becomes the largest contributor to the uncertainty and the share of the internal variability decreases rapidly. However, our uncertainty quantification may erroneously attribute a part of the internal variability of the climate simulations to ESM uncertainty . In order to estimate this error, we forced BESSI with 10 different realizations of the ESM ACCESS-ESM1-5 (Table ) and applied the method of by replacing the different ESMs with the different realizations of a single ESM. This shows a non-negligible bias in the attribution of the uncertainty in the first decades, up to 35 %, adding a caveat to the relative uncertainties in Fig. b for this time period. Note, however, that multiple realizations are available for only about half of the ESMs, and thus we cannot systematically investigate this effect. More importantly for the results of this study, the method bias is small at the end of the century, which means that the ESM uncertainty is robustly shown to be greater than the scenario uncertainty. In other words, in the scenarios with strong forcing, there are some ESMs that induce only small SMB changes, while other ESMs lead to a much stronger SMB decrease. This pronounced uncertainty is larger than the differences between the medians over the ESMs for each scenario. At the end of the century, the ESM uncertainty is about 62 % and the scenario uncertainty is about 35 % of the total variance, whereas the snow model parameter uncertainty and the internal variability represent about 3 % combined.

The separation of variances can be generalized to every grid cell of the GrIS. The total variance of the 1952 simulations is largest at the margin of the ice sheet, where the SMB changes considerably (Fig. a and b). The total variance increases by several orders of magnitude from the middle to the end of the century. At the middle of the century, the ESM uncertainty is the most important component at the margin and in the centre of the ice sheet (Fig. c). Only in the north and at higher altitudes in the west is the internal variability at its largest. Compared to the other components, the scenario uncertainty is insignificant at the middle of the century (Fig. e). At the end of the century, the scenario uncertainty becomes more pronounced, especially at the western margin, where the amount of melt differs considerably between the scenarios (Fig. d). The area where the ESM uncertainty has the largest share increases even more at the end of the century, mainly at the expense of the regions where the internal variability is important at the middle of the century. The scenario uncertainty is of similar magnitude as the ESM uncertainty only at the margins of the ice sheet and in the area where the total variance is low.

In the single-forcing simulations, we run the snow model using only one input variable from each CMIP model simulation. This variable is hereafter called the transient variable. For the other variables, daily means of the historical period of ERA-Interim data are used in the simulation, except for precipitation, whose temporal distribution is again adapted as described in Sect. . We study the influence of the different input variables on the SMB across the entire GrIS and show three regions previously used by (Fig. ). These regions are selected because they illustrate the spatial differences in the behaviour of the SMB.

The SMB increases when precipitation is the transient variable due to an increase in snowfall (Fig. c). In the simulation with transient dew point, the SMB also increases through an increase in desublimation, but the effect is smaller. When the downwelling longwave radiation increases, the snow temperature rises, which leads to more melt. The effect of melting caused by increased air temperature is stronger than that of increased longwave radiation except for the east where the SMB change is dominated by precipitation changes. The interquartile range is largest when temperature or precipitation are the transient variables, except for the east where the dew point has a larger interquartile range than the temperature. Consequently, these variables dominate the uncertainty of the SMB simulations. Shortwave radiation alone has a negligible influence on the SMB in the idealized experiments performed. This does not agree with , who found a link between amplified melt and recent increases in shortwave radiation through shifts in North Atlantic Oscillation and Greenland blocking. However, the ESMs used in this study predict a decrease in shortwave radiation, which could explain the disagreement. In addition, Greenland blocking is not well represented in ESMs .

The sum of all individual changes does not equal the fully transient simulation driven by the SSP585 scenario (Fig. ). This highlights non-linearities that amplify the SMB reduction. For example, air temperature and precipitation often covary so that the increased precipitation compensates the increased melt only partly. If heavier precipitation delivers more rain, the energy required to refreeze the additional rain in the snowpack increases its heat. We conclude that when air temperature and longwave radiation rise together in a warmer and cloudier future and more energy is available at the surface and due to the non-linearity of the SMB, increased melt is detected than from each of these forcing components individually. The impact of the increasing amount of longwave radiation decreases with rising surface temperature because the net flow of sensible heat depends on the temperature difference between air and snow surface. Since the sublimation is driven by the saturation pressure difference between the lower atmosphere and the surface, sublimation increases for a warmer surface and decreases for a higher dew point temperature. In the different ESMs, the SMB reduction is amplified to different extents by the described non-linear effects. Therefore, the interquartile range in the fully transient simulation is larger than the interquartile range in each of the single-forcing simulations.

In the western region, the SMB and its different components follow a similar course as for the entire GrIS, except for the amount of SMB decrease per area, which is in the fully transient simulation about 5 times as high (Fig. ). Additionally, the internal variability is not as important as in the total GrIS, and the scenario uncertainty is slightly higher (Fig. a). This shows a high dependence of surface melt on the climate scenario in this region.

In the northern region, the SMB increases with transient precipitation and transient dew point to the same extent (Fig. c). Desublimation and sublimation are important contributors to the SMB in this dry region. This is in line with , who show that 28 % of the accumulation is caused by desublimation at one station in the northeast at 2113 m above sea level. Even the precipitation increase will not dominate in the north by the end of the century. In the fully transient simulation and in the simulation with transient temperature, the SMB decreases strongly and non-linearly at the end of the century (Fig. c, orange line). The decrease in SMB is rather late because of the low temperatures in the north at present day. However, when the temperatures rise high enough, ice can be exposed at the surface, which is not always covered by the scarce snowfall and thus triggers a strong albedo feedback. The uncertainty associated with the choice of ESM has a larger share in the north than across all of Greenland because the temperature differences between ESMs are more pronounced, which suggests discrepancies in the simulated sea ice cover. As a consequence, the scenario uncertainty is reduced (Fig. b).

In the east, the SMB with transient precipitation follows the SMB with all variables transient closely, showing that the main cause for SMB changes is the precipitation (Fig. d). also found increased precipitation in the east because the reduced sea ice cover leads to a moister atmosphere. The uncertainty ranges between ESMs for transient precipitation and the fully transient simulation are also very similar; therefore, the ESM uncertainty is mostly a precipitation uncertainty. The internal variability has a large contribution to uncertainty (Fig. c) because the total uncertainty of all other components is small (not shown). The ESM uncertainty is still the largest component, showing an increase at the end of the century (Fig. c) when the fully transient SMB stagnates (Fig. d).